Noncommutative knot theory

Abstract

The classical abelian invariants of a knot are the Alexander module, which is the first homology group of the the unique infinite cyclic covering space of S^3-K, considered as a module over the (commutative) Laurent polynomial ring, and the Blanchfield linking pairing defined on this module. From the perspective of the knot group, G, these invariants reflect the structure of G^(1)/G^(2) as a module over G/G^(1) (here G^(n) is the n-th term of the derived series of G). Hence any phenomenon associated to G^(2) is invisible to abelian invariants. This paper begins the systematic study of invariants associated to solvable covering spaces of knot exteriors, in particular the study of what we call the n-th higher-order Alexander module, G^(n+1)/G^(n+2), considered as a Z[G/G^(n+1)$-module. We show that these modules share almost all of the properties of the classical Alexander module. They are torsion modules with higher-order Alexander polynomials whose degrees give lower bounds for the knot genus. The modules have presentation matrices derived either from a group presentation or from a Seifert surface. They admit higher-order linking forms exhibiting self-duality. There are applications to estimating knot genus and to detecting fibered, prime and alternating knots. There are also surprising applications to detecting symplectic structures on 4-manifolds. These modules are similar to but different from those considered by the author, Kent Orr and Peter Teichner and are special cases of the modules considered subsequently by Shelly Harvey for arbitrary 3-manifolds.

Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-19.abs.html

References in corpus (1)

• Approximating L2-invariants, and the Atiyah conjecture

Cited by in corpus (28)

• Knot Concordance and Higher-Order Blanchfield Duality
• Homology Cobordism Invariants and the Cochran-Orr-Teichner Filtration of the Link Concordance Group
• Primary decomposition and the fractal nature of knot concordance
• Twisting $L^2$-invariants with finite-dimensional representations
• Homology and derived series of groups
• New topologically slice knots
• The Magnus representation and higher-order Alexander invariants for homology cobordisms of surfaces
• Homology and Derived p-Series of Groups
• Link concordance and generalized doubling operators
• Universal $L^2$-torsion, polytopes and applications to $3$-manifolds
• Derivatives of Knots and Second-order Signatures
• The $L^2$-Alexander torsion is symmetric
• $L^2$-Euler characteristics and the Thurston norm
• Higher-order signature cocycles for subgroups of mapping class groups and homology cylinders
• Splittings of von Neumann rho-invariants of knots
• Von Neumann rho invariants and torsion in the topological knot concordance group
• Homology cylinders of higher-order
• Homotopy ribbon concordance, Blanchfield pairings, and twisted Alexander polynomials
• L^2-Betti numbers of hypersurface complements
• A colored operad for string link infection
• Thurston norm via Fox calculus
• Twisted Blanchfield pairings and twisted signatures II: Relation to Casson-Gordon invariants
• The First-Order Genus of a Knot
• Linear independence of cables in the knot concordance group
• Genus generators and the positivity of the signature
• An explicit relation between knot groups in lens spaces and those in $S^3$
• $\mathcal{Z}$-compactifiable manifolds which are not pseudo-collarable
• Amenable signatures, algebraic solutions, and filtrations of the knot concordance group